Finite-Dimensional Reduction, Pfaffian Systems, and Creative Telescoping

We develop a unified algebraic framework showing that finite-dimensional reduction of Laplace-type integrals simultaneously yields Pfaffian systems, Gauss-Manin connections, and creative telescoping relations. Building on the reduction quotient formalism introduced in a companion work, we study amplitudes modulo reduction-exact terms and show that parameter differentiation induces well-defined operators on the resulting quotient. Our main result proves that if the class of an amplitude lies in a finite-dimensional subspace of the reduction quotient that is invariant under the descended twisted parameter operators, then the associated family of Laplace-type integrals satisfies a finite-rank Pfaffian system whose connection matrices arise from the induced action of those operators. As a consequence, the corresponding period vector is governed by finite-dimensional linear algebra on the quotient, and scalar differential operators (telescopers) annihilating the integral are obtained from linear dependence in that invariant finite-dimensional subspace. We further establish a normal form theory reducing arbitrary amplitudes to a finite set of master periods, derive explicit elimination identities from iterated reduction, and identify the resulting Pfaffian systems with Gauss-Manin connections on twisted cohomology. Finally, we classify the one-dimensional case: finite-dimensional reduction occurs precisely when the amplitude becomes exponential-polynomial in the phase variable, yielding a complete characterization of finite-rank reduction pairs. These results provide a conceptual and structural explanation for the emergence of holonomicity and creative telescoping in Laplace-type integrals, replacing algorithmic constructions with a finite-dimensional reduction principle.